In an exchange economy with

Two persons (A and B)

Two goods (1 and 2)

Where

p denotes price

x(p) denotes demand function

w denots endownment

e(p) denotes net demand function for each individual

z(p) denotes excess demand function for the market

Here goes the prove for Walras's Law for 2 market case (Varian Intermediate Microeconomics)

First consider two identities:

p1 * xA1(p1,p2) + p2 * xA2(p1,p2) = p1 * wA1 + p2 * wA2 (A's budget constraint)

p1 * xB1(p1,p2) + p2 * xB2(p1,p2) = p1 * wB1 + p2 * wB2 (B's budget constraint)

Rearrange:

p1 [xA1 - wA1] + p2 [xA2 - wA2] =0 => p1 * eA1(p1,p2) + p2 * eA2(p1,p2) = 0

p1 [xB1 - wB1] + p2 [xB2 - wB2] =0 => p1 * eB1(p1,p2) + p2 * eB2(p1,p2) = 0

Adding two equation together:

p1 [eA1 + eB1] + p2 [eA2 + eB2] = 0 =>

p1 * z1(p1,p2) + p2 * z2(p1,p2) = 0

Therefore, if N-1 market (in this case where N=2) is in equilibrium:

z1(p1,p2) = 0

Then the remaining market must in equilibrium:

z2(p1,p2) = 0